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Physics clearly needs more than the bare homotopy types of HoTT. Field configurations may be groupoids (1-types) under gauge equivalence, or indeed ∞−groupoids\infty-groupoids (homotopy types) under gauge-of-gauge-of-… equivalence, but they also possess differentiable structure. The question then is how to cater for all of those principal bundles, connections, curvature forms, and in more recent times 2-bundles, orbifolds, Lie ∞\infty-algebroids,…, while building on HoTT, or, in terms of the environments in which HoTT (or Univalent Foundations) functions, while adding structure to (∞,1)(\infty, 1)-toposes.

October 21, 2013

Jet Categories at the nForum

Posted by David Corfield

Some people I talk to who have noticed a slackening off at the Café in recent months, and who know that some of this is due to John’s energy passing to his Azimuth project, don’t seem aware that another chunk of the energy didn’t just vanish, but got transmitted to the nForum. This venue has the advantage of democratically allowing anyone to initiate a discussion, but the disadvantage that people don’t seem to want to wade through every announcement of any alteration to an nLab entry for the occasional interesting nugget. For whatever reason, we don’t get visited much nowadays by some of the prestigious visitors of yesteryear. Still, if you want to see the day to day movements of Urs sweeping up great clumps of mathematical physics into a glorious synthetic package, the nForum is the place to be. Or, if you prefer to read the finished product, see his site.

Something we occasionally suggest to each other is a periodic digest of what’s happening at the nnLab, but I believe we’ve only managed two to date (I, II). Let me try something less ambitious.

October 16, 2013

Announcing the Kan Extension Seminar

Posted by Emily Riehl

Daniel Kan’s influence at MIT persists through something called the Kan seminar, a graduate reading course in algebraic topology. Over the course of a semester, each student is asked to give a few one-hour lectures summarizing classic papers in the field and to engage with each other paper by writing a reading response. The lectures are preceded by a practice talk of unbounded length that is conducted in private, i.e., in the absence of the lead instructor, before the reading responses are due. This format aims to teach students how to read papers quickly and at various levels of depth, as well as to work on presentation skills. At the semester’s conclusion, Kan traditionally hosted a party that took advantage of Boston’s high concentration of mathematicians, giving his students an opportunity to meet senior people in the field.

This (northern hemisphere) spring, from early January to late June 2014, I plan to run an online (“extension”) Kan seminar in category theory with the aim of reading the twelve papers listed below. I am seeking between 6 and 12 participants who will compose one or two blog posts to appear here on the nn-Category Café over the course of the six months, which will be published every other week. Everyone will be expected to write comments, engaging with all of the papers.

Lawvere, An elementary theory of the category of sets

Street, The formal theory of monads

Freyd-Kelly, Categories of continuous functors, I

Lawvere, Metric spaces, generalized logic and closed categories

Kelly-Street, Review of the elements of 2-categories

Street-Walters, Yoneda structures on 2-categories

Johnstone, On a topological topos

Kelly, Elementary observations on 2-categorical limits

Blackwell-Kelly-Power, Two-dimensional monad theory

Adámek-Borceux-Lack-Rosický, A classification of accessible categories

October 6, 2013

Unexpected Connections

Posted by Tom Leinster

On Wednesday I’ll give a half-hour talk to all the new maths PhD students in Scotland, called Unexpected connections. What should I put in it?

When British students arrive to do a PhD, they have already chosen a supervisor, and they have a fairly good idea of what their PhD topic will be. So, they’re in the mood to specialize. On the other hand, they’re obliged to take some courses, and here in Scotland the emphasis is on broadening — balancing that specialization by learning a wide range of subjects.

Some students don’t like this. They’re not undergraduates any more, they’ve decided what they want to work on, and they resent being made to study other things. My job is to enthuse them about the wider, wilder possibilities — to tell them about some of the amazing advances that have been made by bringing together parts of mathematics that might appear to be completely unconnected.

What are some compelling stories I could tell? What are your favourite examples of apparently disparate mathematical topics that have been brought together to extraordinary effect?

The more disconnected the topics seem to be, the better. Best of all would be stories that connect pure mathematics with either applied mathematics or statistics.